LIBRARY 


UNIVERSITY  OF  CALIFORNIA 


SYLL AB  U  S 

PLANE 
GEOMETRY 

CORA   L.    WILLIAMS 


SYLLABUS   O  F 
PLANE   GEOMETRY 

(  Corresponding  to  Euclid,  Book  I  -  VI ) 


or  THE 
UNIVERSITY 

OF 


PREPARED  AS  AN  INTRODUCTION 
TO  ABSOLUTE  GEOMETRY  BY 
CORA  L.  WILLIAMS,  M.  S. 

' 


Copyrighted,  1905 
By  CORA  L.  WILLIAMS 


Standard  Press,  Berkeley 


PREFACE 

HPHE  logical  order  in  geometry  has  held  the  atten- 
1  tion  of  mathematicians  for  more  than  two 
thousand  years,  but  no  essential  progress  was 
made  towards  its  solution,  after  the  time  of  Euclid, 
until  the  nineteenth  century  witnessed  the  creation 
and  development  of  the  non-Euclidean  geometry.  A 
radically  new  departure  was  taken  when  Lobats- 
chewzky  and  Bolyai  demonstrated  that  a  consistent 
geometry  could  be  constructed  under  an  assumption 
which  contradicts  the  parallel  axiom.  This  took  place 
before  the  year  1830,  yet  no  serious  attempt  was  made 
to  reclassify  the  subject  matter  of  geometry  until  the 
close  of  the  century.  The  present  essay,  prepared  as 
a  thesis  for  the  master's  degree  in  the  year  1898,  at- 
tempts a  classification  based  upon  the  three  hypotheses 
which  may  be  made  with  reference  to  the  existence 
of  parallels.  It  makes  no  use  of  the  other  independent 
postulates  of  geometry  enunciated  for  the  first  time  by 
Hilbert  in  the  year  1899.  A  more  minute  classification 
than  the  one  here  presented  is  therefore  possible,  but 
what  order  is  best  adapted  to  the  systematic  unfolding 
of  geometry  for  the  beginner  has  not  yet  been  deter- 
mined. Meanwhile  we  have  at  hand  in  this  essay  a 
working  plan  which  may  be  used  to  good  effect,  before 
the  study  of  geometry  is  abandoned  by  the  pupil,  in 
exhibiting  its  logical  structure.  A  course  of  geometry 
without  some  such  syllabus  is  like  an  arch  without 
a  keystone;  it  is  certain  to  fall  into  fragments. 

IRVING  STRINGHAM. 


TABLE  OF  CONTENTS 
DEFINITIONS  5 

BOOK    I. 

POSTULATE  1 — Any  figure  may  be  moved  from  one 
position  in  space  to  another,  without  change 
of  size  or  shape. 

SECTION  1 — The    Straight   Line  9 

SECTION  2— The  Circle  17 

SECTION  3 — Proportion  20 

BOOK   II. 

POSTULATE  2— Through  a,  given  point  there  is  at 

least  one  straight  line  which  does  not  cut  a 

given  straight  line,  however  far  produced. 

SECTION  1 — The   Straight   Line  25 

SECTION  2— The  Circle  27 

BOOK  III. 

POSTULATE  3 — Through  a  given  point  there  cannot 

~be  more  than  one  straight  parallel  to  a  given 
straight  line. 

SECTION  1— The    Straight   Line  32 

SECTION  2 — Equality  of  Polygons  35 

SECTION  3— The  Circle  38 

SECTION  4 — Proportion  40 


Syllabus  oF  Plane  Geometry 


DEFINITIONS 

Def.  1.  Any  physical  object  takes  up  room.  The 
room  occupied  by  an  object,  considered 
apart  from  the  object,  is  called  a  solid. 

Def.  2.  That  which  separates  a  solid  from  any 
part  of  the  room  surrounding  it,  but  itself 
no  solid,  is  called  a  surface. 

Def.  3.  That  which  separates  one  part  of  a  sur- 
face from  an  adjacent  part  is  called  a  line. 

Def.  4.  That  which  separates  one  part  of  u  line 
from  an  adjacent  part  is  called  a  point. 

A  point  is  not  divisible;  it  has  position 
only. 

The  intersection  of  two  lines  is  a  point. 
The  intersection  of  a  surface  and  a  line  is  a 
point.  The  intersection  of  two  surfaces  is 
a  line. 

Def.  5.  Points,  lines,  surfaces,  or  solids,  or  any 
combination  of  them,  are  called  figures. 

Def.  6.  Any  assumption  concerning  the  relations 
of  figures  to  one  another  is  called  a  postu- 
late.* 


*  See  Helmholtz:     The  Origin  and  Significance  of  Geo- 
metrical Axioms,  Popular  Scientific  Lectures. 


6          SYLLABUS    OF   PLANE    GEOMETRY. 

Def.  7.  When  a  series  of  figures  existing  in 
accordance  with  certain  given  postulates 
form  only  other  figures  belonging  to  the 
same  series,  they  are  said  to  constitute  a 
geometry. 

A  geometry  can  yield  no  relation  be- 
tween its  figures  that  is  not  already  con- 
tained in  its  postulates,  but  the  statement 
of  certain  relations  is  essential  to  the  com- 
plete determination  of  its  figures.  Such 
statements,  capable  of  being  established 
from  previous  assumptions,  are  called  the- 
orems. These  previous  assumptions  may, 
themselves,  be  theorems  or  postulates. 

It  is  evident  that  only  those  theorems  are 
fundamental  to  a  geometry  which  are  re- 
quired to  establish  its  integrity.  The  other 
theorems,  practically  unlimited  in  number, 
express  relations  between  figures  in  gen- 
eral. 

Def.  8.  The  figure  which  contains  the  figures,  as 
they  exist  in  accordance  with  the  postulates 
of  a  geometry,  is  called  the  space,  or  space 
form  of  that  geometry. 

The  effect  of  introducing  into  a  geometry 
a  new  postulate  is  to  define  more  completely 
the  character  of  its  space-form. 

Any  postulate  may  be  introduced  into  a 


SYLLABUS    OF   PLANE    GEOMETRY.          7 

geometry  which  does  not  destroy  the  integ- 
rity of  the  geometry. 

Def.  9.  The  space-form  of  a  geometry  is  com- 
pletely determined  when  no  postulate  can 
be  added  to  the  geometry  which  is  not  an 
equivalent  of  a  postulate  already  included 
in  it. 

Def.  10.  When  the  space-form  of  a  geometry  is 
completely  determined,  the  geometry  may  be 
said  to  be  logically  complete. 

Def.  11.  The  geometries  which  are  logically  com- 
plete for  the  same  series  of  figures  together 
constitute  what  is  called  the  absolute  geom- 
etry of  .those  figures. 


PLANE  GEOMETRY 
BOOK    I. 

Post.  1.*  Any  figure  may  be  moved  from  one  posi- 
tion in  space  to  another,  without  change 
of  size  or  shape. 

Figures  may  be  compared  with  respect 
to  extent,  or  size,  and  shape.  Figures 
which  may  be  considered  as  extending  in 
either  of  two  opposite  directions  may,  also, 
be  compared  in  respect  to  sense.  The  sense 
of  the  figure  is  called  positive  when  the 
figure  is  thought  to  extend  in  one  direction 
and  negative  in  the  opposite  direction.  It 
is  optional  which  of  the  two  opposite  direc- 
tions is  to  be  regarded  as  positive,  and 
which  negative.  But,  the  selection  once 
made,  that  convention  must  be  observed 
throughout  the  entire  case. 

The  sense  of  figures  is  usually  disre- 
garded except  where  its  introduction  is 
necessary  for  a  complete  generalization  of 
the  theorem. 

Def.  12.  If  two  figures  are  so  related  that  points 
in  the  one  take  the  place  of  lines  in  the 


*  For  a  full  discussion  of  this  postulate,  see  Clifford's 
lectures  on  the  "Philosophy  of  the  Pure  Sciences,"  Lectures 
and  Essays  (p.  222.) 


SYLLABUS    OF   PLANE    GEOMETRY.          9 

other  and  lines  in  the  one,  of  points  in  the 
other,  the  two  figures  are  called  reciprocal 
figures  and  corresponding  theorems  con- 
cerning the  two  figures,  reciprocal  theor- 
ems.* 

SECTION  I. 
The  Straight  Line, 

Def.  13.  If  a  line  is  uniquely  determined  by  any 
two  points  within  it,  which  have  no  special 
relation  to  one  another,  it  is  called 
a  straight  line. 

Def.  14.  Any  portion  of  a  straight  line  cut  off 
by  two  fixed  points  is  called  a  segment. 

In  certain  spaces  there  may  be  points  so 
related  that  any  number  of  straight  lines 
can  be  drawn  between  two  of  them,  as  in 
the  case  of  the  great  circles  joining  the 
poles  of  a  sphere.  But  these  are  exceptional 
cases  occurring  only  when  the  points  bear 
a  special  relation  to  one  another.  In  gen- 
eral, one  straight  line,  and  only  one,  can  be 
drawn  through  two  points. 

Def.  15.  If  a  surface  is  uniquely  determined  by 
any  three  points  within  it,  which  have  no 


*  Olaus  Henrici:     Elementary  Geometry,  Congruent  Fig- 
ures. 


10        SYLLABUS    OF   PLANE    GEOMETRY. 

special  relation  to  one  another,  it  is  called 
a  plane  surface,  or  plane.* 

Def.  16.  A  figure  formed  by  straight  lines  is 
called  a  rectilineal  figure.  A  rectilineal  fig- 
ure of  one  straight  line  is  called  an  uni- 
lateral; of  two  straight  lines,  a  bilateral; 
of  three  straight  lines  a  trilateral ;  of  four 
straight  lines,  a  quadrilateral;  etc. 

Def.  17.  The  figure  which  has  for  its  reciprocal,  a 
segment  is  called  an  angle. 

A  segment  is  determined  by  two  points, 
its  extremities  lying  in  a  straight  line; 
hence  an  angle  must  be  determined  by  two 
lines  meeting  in  a  point.  The  lines  are 
called  the  sides  of  the  angle  and  their  com- 
mon point,  the  vertex. 

As  a  segment  may  be  formed  by  the  move- 
ment of  a  point  from  one  position  in  a 
straight  line  to  another,  an  angle  may  be 
formed  by  the  movement  of  a  straight  line 
about  a  point  in  it  from  one  position  to 
another.  The  size  of  the  segment  depends 
upon  the  amount  of  movement  of  the  point, 
likewise  the  size  of  the  angle  depends  upon 
the  amount  of  turning  of  the  line  and  is 


*  In  the  theorems  and  propositions  which  follow  the 
words,  "in  the  same  plane,"  are  to  be  understood  when  any 
other  possibility  might  arise. 


SYLLABUS    OF   PLANE    GEOMETRY.        11 

entirely  independent  of  the  length  of  the  line 
or  the  extent  of  surface  passed  over. 
Def.  18.  To  fix  that  sense  of  an  angle  which  shalJ 
be  regarded  as  positive,  the  rotation  in  a 
direction  opposite  to  that  of  the  hands  of 
a  watch  is  called  positive. 

No  cognizance  is  taken  of  angles  formed 
by  the  turning  of  a  line  through  more  than 
one  complete  rotation. 

Def.  19.  Two  angles  are  said  to  be  equal  if  the 
bilaterals  forming  them  can  be  brought  into 
coincidence  in  such  a  way  that  the  rotating 
lines  pass  over  the  same  surface. 

Def.  20.  The  bisector  of  an  angle  is  the  straight 
line  that  divides  it  into  two  equal  angles. 

Def.  21.  An  angle  is  said  to  be  a  straight  angle 
or  a  perigon,  according  as  its  arms  lie  in  the 
same  straight  line  on  opposite  sides  or  on 
the  same  side  of  the  vertex. 

Def.  22.  Angles  having  the  same  vertex  and  a  com- 
mon arm  are  called  adjacent  angles.* 

Def.  23.  The  angle  formed  by  a  line  rotating  from 
one  exterior  arm  to  the  other  and  passing 
through  the  common  arm  is  called  the  sum 
of  the  two  adjacent  angles 

Def.    24.        When  two  straight  lines  intersect  so  as 


*  Halsted:     Elementary  Synthetic  Geometry. 


12        SYLLABUS    OF   PLANE    GEOMETRY. 

to  make  two  of  the  adjacent  angles  equal, 

each  of  the  angles  formed  is  called  a  right 

angle. 
Def.    25.        A  perpendicular  to  a  straight  line  is  a 

straight  line  that  makes  a  right  angle  with 

it. 
Def.    26.        An  angle  less  than  a  right  angle  is  said  to 

be  acute;  an  angle  greater  than  a  right 

angle  but  less  than  a  straight  angle  is  said 

to    be    obtuse;    an    angle    greater   than    a 

straight  angle  but  less  than  a  perigon  is 

said  to  be  reflex. 
Def.    27.        Two  angles  are  said  to  be  complements 

of  each  other,  or  complemental,  if  their  sum 

is  a  right  angle. 
Def.    28.        Two  angles  are  said  to  be  supplements 

of  each  other,  or  supplemental,  if  their  sum 

is  a  straight  angle. 
Def.   29.        Two   angles   are   said   to   be   conjugates 

of  each  other  if  their  sum  is  a  perigon. 
Def.    30.        The  opposite  angles  of  a  bilateral  are 

called  vertical  angles. 
Theor.  A.*     All  straight  angles  are  equal. 

(By  post.  1  and  the  def.  of  a  straight  line.) 
Cor.      1.        All  right  angles  are  equal. 


*  The  theorems  denoted  by  letters  are  considered  to  be 
fundamental.     See  def.  7.) 


SYLLABUS    OF   PLANE    GEOMETRY.        13 

Cor.  2.  If  a  straight  line  stands  upon  another 
straight  line,  it  makes  the  adjacent  angles 
together  equal  to  two  right  angles. 

Cor.  3.  All  the  angles  made  by  any  number  of 
straight  lines  drawn  from  a  point,  each  with 
the  next  following  in  order,  are  together 
equal  to  four  right  angles. 

COP.  4.  The  complements  of  equal  angles  are 
equal. 

Cor.  5.  The  supplements  of  equal  angles  are 
equal. 

Cor.  6.  The  vertical  angles  of  a  bilateral  are 
equal. 

Prop.  1.*  At  a  given  point  in  a  given  straight  line 
not  more  than  one  perpendicular  can  be 
drawn  to  that  line.  (Indirect  method.) 

Prop.  2.  If  the  adjacent  angles  made  by  one 
straight  line  with  two  others,  all  three 
meeting  at  a  point,  are  together  equal  to 
two  right  angles,  these  two  straight  lines 
are  in  one  straight  line. 

Def.    31.        Two  figures  are  said  to  be  congruent  if 


*  In  addition  to  the  theorems  selected  as  fundamental, 
the  proofs  of  many  others  are  to  be  found  in  the  usual  text- 
books on  Euclid.  Those  here  given  under  the  head  of  propo- 
sitions complete  the  list  as  given  in  the  Syllabus  prepared  by 
the  Association  for  the  Improvement  of  Geometrical  Teach- 
ing. They  are  added  for  the  purpose  of  showing  how  the  sub- 
ject matter  of  geometry  distributes  itself  under  the  plan  here 
presented. 


14        SYLLABUS    OF   PLANE    GEOMETRY. 

they  can  be  made  by  superposition  to  coin- 
cide with  one  another. 

Def.  32.  If  in  a  rectilinear  figure,  points  of  inter- 
section, to  the  number  of  its  lines,  are  so 
selected  that  two,  and  only  two,  points  fall 
on  each  line,  the  figure  formed  by  the  seg- 
ments thus  cut  off  on  the  lines  is  called  a 
polygon. 

The  points  of  intersection  which  deter- 
mine the  polygon  are  called  its  vertices,  and 
the  segments  joining  them,  its  sides. 

The  surfaces  enclosed,  or  the  sum  of  the 
surfaces  enclosed,  is  called  the  area  of  the 
polygon.  (Negative  areas  are  to  be  con- 
sidered. ) 

If  the  sides  of  a  polygon  are  produced 
in  succession,  either  way,  the  angles  deter- 
mined by  the  sides  produced  and  the  follow- 
ing sides  are  called  the  exterior  angles  of 
the  polygon;  the  supplements  of  the  ex- 
terior angles  are  called  the  interior  angles, 
or  simply  angles,  of  the  polygon. 

A  straight  line,  other  than  a  side,  joining 
any  two  vertices  of  a  polygon  is  called  a 
diagonal. 

The  sum  of  the  sides  is  called  the  peri- 
meter of  a  polygoni 

Def.  33.  A  polygon,  no  side  of  which  cuts  another 
when  produced,  is  called  a  convex  polygon. 


SYLLABUS    OF   PLANE    GEOMETRY.        15 

Def.  34.  A  polygon  of  three  angles  is  a  trigon,  or 
triangle;  of  four  angles,  a  tetragon;  of  five 
angles,  a  pentagon;  of  six,  a  hexagon;  etc. 

Theor.  B.  If  two  triangles  have  two  sides  and  the 
included  angle  of  the  one  equal  respectively 
to  two  sides  and  the  included  angle  of  the 
other,  the  triangles  are  congruent. 

(If  the  sides  of  one  triangle  are  arranged 
in  reverse  order  to  the  corresponding  sides 
of  a  second  triangle,  one  of  the  triangles 
must  be  turned  over  before  it  can  be  brought 
into  coincidence  with  the  other.  Now, 
under  certain  conditions  it  may  be  neces- 
sary for  a  proper  understanding  of  the 
problem  to  draw  the  figures  on  a  surface 
other  than  a  plane.  For  example,  if  the 
question  of  the  possible  intersection  of  two 
lines  in  a  second  point  should  arise,  a  more 
suggestive  figure  can  be  obtained  by  using 
the  surface  of  a  sphere.  In  the  turning  of 
such  a  figure  the  surface  must  undergo 
flexure.  This  is  not  true,  however,  of  the 
plane  surface  in  which  the  figure,  thus  rep- 
resented, actually  lies.) 

Theor.  0.  If  two  triangles  have  two  angles  and  the 
included  side  of  the  one  equal  respectively 
to  two  angles  and  the  included  side  of  the 
other,  the  triangles  are  congruent. 


16        SYLLABUS    OF   PLANE    GEOMETRY. 

Def.  35.  If  a  triangle  has  two  equal  sides,  it  is 
called  an  isosceles  triangle. 

Def.  36.  The  line  drawn  from  any  vertex  of  a  tri- 
angle to  the  middle  point  of  the  opposite 
side  is  called  the  median  to  that  side. 

Prop.  3.  In  an  isosceles  triangle,  the  angles  oppo- 
site the  equal  sides  are  equal. 

Prop.  4.  An  equilateral  triangle  is  also  equiangu- 
lar. 

Prop.  5.  If  two  angles  of  a  triangle  are  equal,  the 
triangle  is  isosceles. 

Prop.  6.  An  equiangular  triangle  is  also  equilat- 
eral. 

Prop.  7.  If  two  triangles  have  the  three  sides  of 
the  one  respectively  equal  to  the  three  sides 
of  the  other,  the  triangles  are  congruent. 

Prop.  8.  If  two  triangles  have  two  sides  of  the 
one  equal  respectively  to  two  sides  of  the 
other  and  the  angles  opposite  one  pair  of 
equal  sides  equal,  then  the  angles  opposite 
the  other  pair  of  equal  sides  are  either 
equal  or  supplemental,  and,  if  equal,  the 
triangles  are  congruent. 

Def.  37.  A  line  cutting  two  or  more  lines  is  called 
a  transversal  of  those  lines. 

The  angles  formed  by  the  transversal  with 
either  line  on  the  side  opposite  to  the  other 
are  called  exterior  angles. 

The  angles  formed  by  the  transversal  with 


SYLLABUS    OF   PLANE    GEOMETRY.        17 

either  line  on  the  same  side  as  the  other  are 
called  interior  angles. 

The  interior  angles,  or  the  exterior  angles, 
on  opposite  sides  of  the  transversal  are 
called  alternate  angles. 

An  exterior  angle  and  the  non-adjacent 
interior  angle  on  the  same  side  of  the  trans- 
versal are  called  corresponding  angles. 

Prop.  9.  If  a  transversal  of  two  lines  makes  a  pair 
of  alternate  angles  equal,  then  any  angle 
is  equal  to  its  alternate  angle,  any  angle  is 
equal  to  its  corresponding  angle,  and  any 
two  interior  or  any  two  exterior  angles  on 
the  same  side  of  the  transversal  are  sup- 
plemental. 

Prop.  10.  The  same  conclusions  follow,  if  two  corre- 
sponding angles  are  equal,  or  if  two  interior 
or  two  exterior  angles,  on  the  same  side  of 
the  transversal  are  supplemental. 


SECTION  2. 
The  Circle. 

Def.  38.  The  circle  is  a  portion  of  a  surface 
bounded  by  one  line  called  the  circumfer- 
ence and  is  such  that  all  points  on  that 
line  are  equidistant  from  a  point  within 
the  figure  called  the  center  of  the  circle. 


18        SYLLABUS    OF    PLANE    GEOMETRY. 

Def.  39.  A  straight  line  drawn  from  the  center 
to  the  circumference  is  called  a  radius. 

Def.  40.  A  straight  line  drawn  through  the  center 
and  terminated  both  ways  by  the  circumfer- 
ence is  called  a  diameter. 

Prop.  11.  Circles  having  equal  radii  are  congruent. 
( Superposition. ) 

Prop.  12.  Any  diameter  of  a  circle  divides  it  into 
two  congruent  parts  called  semicircles. 

Prop.  13.  Any  two  diameters  at  right  angles  to  one 
another  divide  the  circle  into  four  congru- 
ent parts  called  quadrants. 

Def.  41.  The  figure  formed  by  all  points  satisfying 
a  given  condition  is  called  the  'locus  of  all 
points  satisfying  that  condition. 

Prop.  14.  The  circumference  of  a  circle  is  the  locus 
of  all  points  at  the  distance  of  the  radius 
from  the  center. 

Def.    42.        An  arc  is  a  part  of  a  circumference. 

Two  arcs  which  together  form  a  circum- 
ference are  said  to  be  conjugate.  The 
greater  of  the  two  arcs  is  called  the  major 
arc,  and  the  smaller,  the  minor  arc. 

Def.  43.  An  angle  at  the  center  is  said  to  stand 
upon  the  arc  which  lies  within  the  angle 
and  is  cut  off  by  the  sides  of  the  angle. 

Def.  44.  A  portion  of  a  circle  cut  off  by  an  arc 
and  two  radii  drawn  to  its  extremities  is 
called  a  sector,  and  the  angle  at  the  center 


SYLLABUS    OF   PLANE    GEOMETRY.        19 

which  stands  upon  that  arc  is  called  the 
angle  of  the  sector. 

Theor.  D.  In  the  same  circle,  or  in  equal  circles,  if 
two  angles  at  the  center  are  equal,  the  arcs 
on  which  they  stand  are  also  equal,  and  of 
two  unequal  angles  at  the  penter,  the  greater 
stands  on  the  greater  arc. 

Prop.  15.  In  the  same  circle,  or  in  equal  circles, 
sectors  which  have  equal  angles  are  equal, 
and  of  two  such  sectors  which  have  unequal 
angles,  the  greater  is  that  which  has  the 
greater  angle. 

Prop.  16.  In  the  same  circle,  or  in  equal  circles,  if 
two  arcs  are  equal,  the  angles  which  they 
subtend  at  the  center  are  also  equal,  and 
of  two  unequal  arcs,  the  greater  subtends 
the  greater  angle  at  the  center. 

Prop.  17.  In  the  same  circle,  or  in  equal  circles, 
equal  sectors  have  equal  angles,  and  of  two 
unequal  sectors,  the  greater  has  the  greater 
angle. 

Def.  45.  The  straight  line  joining *any  two  points 
on  a  circumference  is  called  a  chord. 

Prop.  18.  In  the  same  circle,  or  in  equal  circles, 
equal  arcs  are  subtended  by  equal  chords. 

Prop.  19.  In  the  same  circle,  or  in  e*qual  circles, 
equal  chords  subtend  equal  arcs. 

Prop.  20.        The  straight  line  drawn  from  the  center 


20        SYLLABUS    OF   PLANE    GEOMETRY. 

to  the  middle  point  of  a  chord  is  perpen- 
dicular to  the  chord. 

Prop.  21.  The  straight  line  drawn  perpendicular  to 
a  chord  through  its  middle  point  passes 
through  the  center  of  the  circle. 

Def.  46.  If  all  the  vertices  of  a  polygon  lie  on  a 
circumference,  the  polygon  is  said  to  be 
inscribed  in  the  circle,  and  the  circle  is  said 
to  be  circumscribed  about  the  polygon. 

Def.  47.  A  polygon  is  said  to  be  regular  when 
it  is  both  equilateral  and  equiangular. 

Prop.  22.  If  the  whole  circumference  of  a  circle 
is  divided  into  any  number  of  equal  arcs, 
the  inscribed  polygon  formed  by  the  chords 
of  these  arcs  is  regular.  (Superposition.) 


SECTION  3. 
Proportion.* 

Def.  48.  A  figure  is  called  a  magnitude  when  its 
size  alone  is  considered. 

Def.  49.  Two  figures  are  said  to  be  magnitudes 
of  the  same  kind  when  they  can  be  com- 
pared with  one  another. 

Def.     50.        A   greater   magnitude   is   said   to   be   a 


*  For  a  geometrical  treatment  of  the  subject  of  Propor- 
tion see  Hall  and  Stevens'  Text  Book  of  Euclid's  Elements, 
Book  V. 


SYLLABUS    OF   PLANE    GEOMETRY.        21 

multiple  of  a  less,  when  the  greater  con- 
tains the  less  an  exact  number  of  times. 

The  following  property  of  multiples  is 
assumed  as  axiomatic: 

mA>=or<mB  according  as  A  > 
=  or  <  B,  where  A  and  B  represent  two 
magnitudes  and  m  is  any  integer. 

The  converse  proposition  necessarily  fol- 
lows: 

A  >  =  or  <  B  according  as  m  A  >  =  or 
<  m  B. 

Def.  51.  The  ratio  of  one  magnitude  to  another 
of  the  same  kind  is  the  relation  which  the 
first  bears  to  the  second. 

The  ratio  of  magnitude  A  to  magnitude  B 
is  denoted  thus,  A:  B,  and  A  is  called  the 
antecedent,  B,  the  consequent. 

If  A  is  equal  to  B,  the  ratio  A:B  is 
called  a  ratio  of  equality.  If  A  is  greater 
than  B,  the  ratio  A  :B  is  called  a  ratio  of 
greater  inequality.  If  A  is  less  than  B, 
the  ratio  A  :B  is  called  a  ratio  of  less  in- 
equality. 

The  ratio  of  A:B  may  be  estimated  by 
examining  how  the  multiples  of  A  are  dis- 
tributed among  the  multiples  of  B  in  their 
relative  scale. 

Def.  52.  The  ratio  of  two  magnitudes  is  said  to  be 
equal  to  a  second  ratio  of  two  other  magni- 


22        SYLLABUS    OF   PLANE    GEOMETRY. 

tudes  (whether  of  the  same  or  of  a  different 
kind  from  the  former),  when  the  multiples 
of  the  antecendent  of  the  first  ratio  are  dis- 
tributed among  those  of  its  consequent  in 
the  same  order  as  the  multiples  of  the  an- 
tecedent of  the  second  ratio  among  those  of 
its  consequent. 

Def.  53.  When  the  ratio  of  A  to  B  is  equal  to  that 
of  P  to  Q  the  four  magnitudes,  A,  B,  P,  Q, 
are  said  to  be  proportionals,  or  to  form  a 
proportion.  The  proportion  is  denoted  thus : 

A:  B   ::  P:  Q, 

which  is  read  A  is  to  B  as  P  is  to  Q.  A  and 
Q  are  called  the  extremes;  B  and  P,  the 
means. 

Def.  54.  The  ratio  of  one  magnitude  to  another 
is  greater  that  that  of  a  third  magnitude  to 
a  fourth,  when  it  is  possible  to  find  equi- 
multiples of  the  antecedents  and  equimul- 
tiples of  the  consequents,  such  that  while  the 
multiple  of  the  antecedent  of  the  first  ratio 
is  greater  than,  or  equal  to,  that  of  its  con- 
sequent, the  multiple  of  the  antecedent  of 
the  second  ratio  is  not  greater  or  is  less, 
than  that  of  its  consequent. 

Def.  55.  Two  ratios  are  said  to  be  reciprocal  when 
the  antecedent  and  the  consequent  of  one 
are  the  consequent  and  the  antecedent  of 
the  other,  respectively. 


SYLLABUS    OF   PLANE    GEOMETRY.        23 

Theor.  E.  Ratios  which  are  equal  to  the  same  ratio 
are  equal  to  one  another. 

Prop.  23.  If  two  ratios  are  equal,  their  reciprocal 
ratios  are  equal. 

Prop.  24.  Equal  magnitudes  have  the  same  ratio  to 
the  same  magnitude;  and  the  same  magni- 
tude has  the  same  ratio  to  equal  magnitudes. 

Prop.  25.  Of  two  unequal  magnitudes,  the  greater 
has  a  greater  ratio  to  a  third  magnitude 
than  the  less  has;  and  the  same  magnitude 
has  a  greater  ratio  to  the  less  of  two  magni- 
tudes than  it  has  to  the  greater. 

Prop.  26.  Magnitudes  which  have  the  same  ratio  to 
the  same  magnitude  are  equal  to  one  an- 
other; and  those  to  which  the  same  magni- 
tude has  the  same  ratio  are  equal  to  one 
another. 

Prop.  27.  That  magnitude  which  has  a  greater  ratio 
than  another  has  to  the  same  magnitude  is 
the  greater  of  the  two ,  and  that  magnitude 
to  which  the  same  magnitude  has  a  greater 
ratio  than  it  has  to  another  magnitude  is 
the  less  of  the  two. 

Prop.  28.  Magnitudes  have  the  same  ratio  one  to 
another  which  their  equimultiples  have. 

Prop.  29.  If  four  magnitudes  of  the  same  kind  are 
proportionals,  the  first  is  greater  than, 
equal  to,  or  less  than,  the  third  according  as 


24        SYLLABUS    OF   PLANE    GEOMETRY. 

the  second  is  greater  than,  equal  to,  or  less 
than,  the  fourth. 

Prop.  30.  If  four  magnitudes  of  the  same  kind  are 
proportionals,  the  first  will  have  to  the  third 
the  same  ratio  as  the  second  to  the  fourth. 
(Alternando). 

Theor.  F.  In  the  same  circle,  or  in  equal  circles,  an- 
gles at  the  center  and  sectors  are  propor- 
tional to  the  arcs  on  which  they  stand. 


SYLLABUS    OF   PLANE    GEOMETRY.        25 

BOOK  II. 

Post  2.  Through  a  given  point  there  is  at  least 
one  straight  line  which  does  not  cut  a  given 
straight  line  however  far  produced.* 

SECTION  1. 
The  Straight  Line. 

Theor.  G.      If  any  side  of  a  triangle  is  produced,  the 

exterior  angle  is  greater  than  either  of  the 

interior  angles  not  adjacent  to  it. 
Prop.  31.        Any  two  angles  of  a  triangle  are  together 

less  than  two  right  angles. 
Prop.  32.        If  a  triangle  has  one  right  angle,  or  one 

obtuse  angle,  its  remaining  angles  are  acute. 
Prop.  33.        From  a  point  outside  a  given  line  not 

more  than  one  perpendicular  can  be  drawn 

to  that  line. 
Def.     56.        A  triangle,  one  of  whose  angles  is  a  right 

angle,  is  called  a  right-angled  triangle. 
Def.     57.        A  triangle,  one  of  whose  angles  is  an 

obtuse   angle,    is   called   an    obtuse-angled 

triangle. 
Def.    58.        A  triangle,  all  of  whose  angles  are  acute, 

is  called  an  acute-angled  triangle. 


*  The  assumption  here  that  space  is  infinite  precludes 
the  possibility  of  two  straight  lines  intersecting  more  than 
once,  also  the  return  unto  itself  of  an  unlimited  straight  line. 


26        SYLLABUS    OF   PLANE    GEOMETRY. 

Def.  59.  The  side  opposite  the  right  angle  of  a 
right-angled  triangle  is  called  the  hypote- 
muse. 

Prop.  34.  If  two  sides  of  a  triangle  are  unequal,  tho 
greater  side  has  the  greater  angle  opposite 
to  it. 

Prop.  35.  If  two  angles  of  a  triangle  are  unequal, 
the  greater  angle  has  the  greater  side  oppo- 
site to  it. 

Prop.  36.  Of  all  the  straight  lines  that  can  be  drawn 
to  a  given  straight  line  from  a  given  point 
outside  it,  the  perpendicular  is  the  short- 
est; of  others,  those  making  equal  angles 
with  the  perpendicular  are  equal;  and  of 
two  others,  that  which  makes  the  greater 
angle  with  the  perpendicular  is  the  greater. 

Prop.  37.  Not  more  than  two  equal  oblique  lines  can 
be  drawn  from  a  given  point  to  a  given 
straight  line. 

Prop.  38.  The  sum  of  any  two  sides  of  a  triangle  is 
greater  than  the  third  side. 

Prop.  39.  The  difference  of  any  two  sides  of  a  tri- 
angle is  less  than  the  third  side. 

Prop.  40.  If  from  a  point  within  a  triangle,  two 
lines  are  drawn  to  the  extremities  of  a  side, 
their  sum  is  less  than  the  sum  of  the  other 
two  sides  of  -the  triangle,  but  they  contain 
the  greater  angle. 

Prop.  41.        If  two  triangles  have  two  sides  of  the  one 


SYLLABUS   OF   PLANE    GEOMETRY.        27 

equal  respectively  to  two  sides  of  the  other, 
but  the  included  angles  unequal,  then  the 
third  sides  are  unequal,  the  greater  side 
being  opposite  the  greater  angle. 

Prop.  42.  If  two  triangles  have  two  sides  of  the  one 
equal  respectively  to  two  sides  of  the  other, 
but  the  third  sides  unequal,  then  the  includ- 
ed angles  are  unequal,  the  greater  angle 
being  opposite  the  greater  third  side. 

Prop.  43.  If  two  triangles  have  two  angles  of  the 
one  equal  respectively  to  two  angles  of  the 
other,  and  the  sides  opposite  one  pair  of 
equal  angles  equal,  the  triangles  are  con- 
gruent. 

Theor.  H.  If  a  transversal  of  two  straight  lines 
make  a  pair  of  alternate  angles  equal,  the 
two  straight  lines  will  not  meet. 

Prop.  44.  If  a  transversal  of  two  straight  lines 
makes  a  pair  of  corresponding  angles  equal, 
or  a  pair  of  interior  angles  on  the  same  side 
supplemental,  the  straight  lines  will  not 
meet. 

SECTION  2. 
The  Circle. 

Prop.  45.        A  diameter  perpendicular  to  a  chord  bi- 
sects the  chord  and  its  subtended  arc. 
Prop.  46.        All  points  in  a  chord  lie  within  a  circle, 


28        SYLLABUS    OF   PLANE    GEOMETRY. 

and  all  points  in  the  chord  produced  lie 
without  the  circle. 

Prop.  47.  A  straight  line  cannot  meet  the  circum- 
ference of  a  circle  in  more  than  two  points. 

Def.  60.  A  straight  line  cutting  the  circumference 
of  a  circle  in  two  points  is  called  a  secant. 

Prop.  48.  Two  circles  whose  circumferences  have 
three  points  in  common  coincide  wholly. 

Prop.  49.  The  circumferences  of  two  circles  which 
do  not  coincide  cannot  meet  in  more  than 
two  points. 

Prop.  50.  If  from  a  point  within  a  circle,  more  than 
two  straight  lines  drawn  to  the  circum- 
ference are  equal,  that  point  is  the  center. 

Prop.  51.  In  the  same  circle,  or  in  equal  circles,  the 
greater  of  two  unequal  minor  arcs  is  sub- 
tended by  the  greater  chord. 

Prop.  52.  In  the  same  circle,  or  in  equal  circles,  if 
two  chords  are  unequal,  the  greater  subtends 
the  greater  minor  and  the  less  major  arc. 

Prop.  53.  In  the  same  circle,  or  in  equal  circles, 
equal  chords  are  equidistant  from  the  cen- 
ter; and  of  two  unequal  chords,  the  greater 
is  nearer  the  center. 

Prop.  54.  In  the  same  circle,  or  in  equal  circles, 
chords  that  are  equidistant  from  the  center 
are  equal ;  and  of  two  chords  unequally  dis- 


SYLLABUS    OF   PLANE    GEOMETRY.        29 

tant,  the  one  nearer  the  center  is  the 
greater. 

Prop.  55.  The  diameter  is  the  greatest  chord  in  a 
circle. 

Prop.  56.  Of  all  straight  lines  passing  through  a 
point  on  the  circumference  of  a  circle,  the 
only  one  that  does  not  meet  the  circumfer- 
ence again  is  perpendicular  to  the  radius  at 
that  point. 

Def.  61.  The  straight  line  which  meets  the  circum- 
ference of  a  circle  in  but  one  point  is  said 
to  touch,  or  be  tangent  to,  the  circle  at  that 
point.  The  point  is  called  the  point  of  con- 
tact or  the  point  of  tangency. 

Prop.  57.  The  center  of  a  circle  lies  in  the  perpen- 
dicular to  any  tangent  at  the  point  of  con- 
tact. 

Prop.  58.  The  radius  perpendicular  to  a  tangent 
meets  it  at  the  point  of  contact. 

Prop.  59.  A  straight  line  cuts,  touches,  or  does  not 
meet  a  circumference,  according  as  its  dis- 
tance from  the  center  is  less  than,  equal  to, 
or  greater  than  the  radius. 

Def.  62.  The  perpendicular  to  a  given  straight 
line  from  a  given  point,  external  to  it,  is 
called  the  distance  of  that  point  from  the 
straight  line. 

Prop.  60.        The  distance  of  a  straight  line  from  the 


30        SYLLABUS    OF   PLANE    GEOMETRY. 

center  of  a  circle  is  less  than,  equal  to,  OP 
greater  than,  the  radius  according  as  the 
straight  line  cuts,  touches,  or  does  not  meet 
the  circumference. 

Def.  63.  Two  circles  are  said  to  touch  or  be  tan- 
gent when  their  circumferences  have  one, 
and  only  one,  point  in  common. 

Prop.  61.  If  two  circumferences  meet  in  a  point 
which  is  not  on  the  line  joining  their  centers, 
they  meet  in  one  other  point ;  their  common 
chord  is  bisected  at  right  angles  by  the  line 
joining  their  centers;  and  the  distance  be- 
tween their  centers  is  greater  than  the  dif- 
ference and  less  than  the  sum  of  the  radii. 

Prop.  62.  If  two  circles  meet  in  one  point  only,  that 
point  lies  in  the  line  joining  their  centers. 

Prop.  63.  If  two  circles  meet  in  a  point  which  is  on 
the  line  joining  their  centers,  they  are  tan- 
gent. 

Prop.  64.  If  two  circles  intersect,  neither  point  of 
intersection  is  on  the  line  joining  their  cen- 
ters. 

Prop.  65.  If  two  circles  touch  one  another,  they  have 
a  common  tangent  line  at  the  point  of  con- 
tact. 

Def.  64.  If  all  the  lines  of  a  rectilineal  figure  are 
tangent  to  a  circle,  the  figure  is  said  to  be 
circumscribed  about  the  circle  and  the  circle 


SYLLABUS    OF   PLANE    GEOMETRY.        31 

is  called  an  inscribed  or  escribed  circle  of 
any  polygon  formed  by  the  figure,  according 
as  it  lies  within  or  without  that  polygon. 
Prop.  66.  If  the  whole  circumference  of  a  circle  is 
divided  into  any  number  of  equal  arcs,  the 
circumscribed  polygon  formed  by  the  tan- 
gents drawn  at  all  the  points  of  division  is 
regular. 


32        SYLLABUS    OF   PLANE    GEOMETRY. 
BOOK  III. 

Def.  65.  Two  straight  lines  which  do  not  meet, 
however  far  produced,  are  said  to  be  paral- 
lel.* 

Post.  3.  Through  a  given  point  there  can  not  be 
more  than  one  straight  line  parallel  to  a 
given  straight  line. 

SECTION  1. 
The  Straight  Line. 

Theor.  I.  The  alternate  angles  made  by  a  trans- 
versal with  two  parallels  are  equal. 

Cor.  The  corresponding  angles  made  by  a 

transversal  with  two  parallels  are  equal. 

Prop.  67.  A  transversal  perpendicular  to  one  of 
two  parallels  is  perpendicular  to  the  other 
also. 

Prop.  68.  The  interior  angles  on  the  same  side  of  a 
transversal  of  two  parallels  are  supplemen- 
tal. 

Prop.  69.  If  the  interior  angles  on  the  same  side 
of  a  transversal  of  two  straight  lines  are 
not  supplemental,  the  straight  lines  are  not 
parallel,  but  meet  on  that  side  of  the  trans- 


*  The   term    "parallel"   thus    defined    includes   the   non- 
intersectors  of  hyperbolic  space. 


SYLLABUS    OF   PLANE    GEOMETRY.        33 

versa  1  on  which  the  sum  of  the  interior  an- 
gles is  less  than  a  straight  angle. 

Theor.  J.  Straight  lines  parallel  to  the  same 
straight  line  are  parallel  to  each  other.* 

Prop.  70.  Any  exterior  angle  of  a  triangle  is  equal 
to  the  sum  of  the  two  opposite  interior 
angles,  and  the  sum  of  the  three  interior 
angles  is  equal  to  two  right  angles. 

Prop.  71.  The  acute  angles  of  a  right-angled  tri- 
angle are  supplemental. 

Prop.  72.  The  sum  of  the  interior  angles  of  a  poly- 
gon is  equal  to  two  right  angles,  taken  as 
many  times  less  two  as  the  figure  has  sides. 

Prop.  73.  The  sum  of  the  exterior  angles  of  a  poly- 
gon is  four  right  angles. 

Def.  66.  A  quadrilateral  whose  opposite  sides  are 
parallel  is  called  a  parallelogram. 

Def.  67.  A  quadrilateral  that  has  one  pair  of  oppo- 
site sides  parallel  is  called  a  trapezoid. 

Prop.  74.  In  a  parallelogram,  any  two  opposite 
angles  are  equal,  and  any  two  consecutive 
angles  are  supplemental. 

Prop.  75.  If  one  angle  of  a  parallelogram  is  a  right 
angle,  all  of  its  angles  are  right  angles. 


*  This  theorem  is  true  for  hyperbolic  space  if  a  distinc- 
tion is  made  between  parallels  (proper)  and  non-intersectors, 
but  it  did  not  seem  advisable  to  make  such  a  distinction  in 
an  Elementary  Geometry. 


34        SYLLABUS    OF   PLANE    GEOMETRY. 

Def.  68.  A  parallelogram,  one  of  whose  angles  is 
a  right  angle,  is  called  a  rectangle. 

Theor.  K.  In  any  parallelogram,  either  diagonal 
divides  it  into  two  congruent  triangles,  also 
the  opposite  sides  are  equal. 

Prop.  76.  If  two  adjacent  sides  of  a  parallelogram 
are  equal,  all  of  its  sides  are  equal. 

Def.  69.  A  rhombus  is  a  parallelogram  that  has 
two  adjacent  sides  equal. 

Def.  70.  A  square  is  a  rectangle  that  has  two  adja- 
cent sides  equal. 

Prop.  77.  If  two  parallelograms  have  two  adjacent 
sides  of  the  one  equal  respectively  to  two 
adjacent  sides  of  the  other,  and  an  angle 
of  the  one  equal  to  an  angle  of  the  other, 
the  parallelograms  are  congruent. 

Prop.  78.  Two  rectangles  are  congruent  if  two  adja- 
cent sides  of  the  one  are  equal  respectively 
to  any  two  adjacent  sides  of  the  other. 

Prop.  79.  Two  squares  are  congruent  if  a  side  of 
the  one  is  equal  to  a  side  of  the  other. 

Prop.  80.  If  a  quadrilateral  has  two  opposite  sides 
equal  and  parallel,  it  is  a  parallelogram. 

Prop.  81.  If  there  are  two  pairs  of  straight  lines  all 
of  which  are  parallel,  and  if  the  intercepts 
made  by  each  pair  on  any  transversal  are 
equal,  then  the  intercepts  on  any  other 
transversal  are  also  equal. 

Prop.  82.        If  a  system  of  parallels  cuts  off  equal 


SYLLABUS    OF   PLANE    GEOMETRY.        35 

segments  on  any  transversal,  it  cuts  off 
equal  segments  on  every  transversal. 

Prop.  83.  The  straight  line  drawn  through  the 
middle  point  of  one  side  of  a  triangle  paral- 
lel to  another  side  bisects  the  third  side. 

Prop.  84.  The  straight  line  joining  the  middle  points 
of  two  sides  of  a  triangle  is  parallel  to  the 
third  side. 

SECTION  2. 
Equality  of  Polygons. 

Def.  71.  The  altitude  of  a  parallelogram  with 
respect  to  a  given  side  as  base  is  the  per- 
pendicular distance  between  the  base  and 
the  opposite  side. 

Def.  72.  The  altitude  of  a  triangle  with  respect 
to  a  given  side  as  base  is  the  perpendicular 
distance  from  the  opposite  vertex  to  the  line 
of  that  base. 

Theor.  L.  Parallelograms  on  the  same  base,  or  on 
equal  bases,  and  between  the  same  parallels 
are  equal. 

Cor.  A  parallelogram  is  equal  to  a  rectangle 

of  the  same  base  and  altitude. 

Prop.  85.  Parallelograms  having  equal  bases  and 
altitudes  are  equal;  and  of  two  parallelo- 
grams having  equal  altitudes,  that  is  the 


36        SYLLABUS    OF    PLANE    GEOMETRY. 

greater  which  has  the  greater  base ;  and  also 
of  two  parallelograms  having  equal  bases, 
that  is  the  greater  which  has  the  greater 
altitude. 

Theor.  M.  Triangles  on  the  same  base,  or  on  equal 
bases,  and  between  the  same  parallels,  are 
equal. 

Cor.  A  triangle  is  equal  to  half  of  a  rectangle 

of  the  same  base  and  altitude. 

Prop.  86.  Triangles  having  equal  bases  and  equal 
altitudes  are  equal. 

Prop.  87.  Equal  triangles  on  the  same  base,  or  on 
equal  bases,  have  equal  altitudes. 

Theor.  N.  Equal  triangles  on  the  same  base,  or  on 
equal  bases  in  the  same  straight  line,  and  on 
the  same  side  of  it,  are  between  the  same 
parallels. 

Prop.  88.  A  trapezoid  is  equal  to  a  rectangle  whose 
base  is  half  the  sum  of  the  two  parallel 
sides,  and  whose  altitude  is  the  perpendicu- 
lar distance  between  them. 

Prop.  89.  The  straight  lines,  drawn  through  any 
point  in  a  diagonal  of  a  parallelogram 
parallel  to  the  sides,  form  equal  parallelo- 
grams on  opposite  sides  of  the  diagonal. 

Def.  73.  All  rectangles  being  congruent  which 
have  two  adjacent  sides  equal  to  two  given 
straight  lines,  any  such  rectangle  is  spoken 
of  as  the  rectangle  of  those  lines. 


SYLLABUS    OF   PLANE    GEOMETRY.        37 

Def.  74.  In  like  manner,  any  square  whose  side 
is  equal  to  a  given  straight  line  is  spoken 
of  as  the  square  of  that  line. 

Def.  75.  A  point  in  a  segment  of  a  straight  line 
is  said  to  divide  it  internally;  a  point  in  a 
segment  produced  is  said  to  divide  it  exter- 
nally. 

Prop.  90.  The  square  of  the  sum  of  two  lines  is 
equal  to  the  sum  of  the  squares  of  those 
lines  and  twice  their  rectangle. 

Prop.  91.  The  square  of  the  difference  of  two  lines 
is  equal  to  the  sum  of  the  squares  of  those 
lines  less  twice  their  rectangle. 

Prop.  92.  The  difference  of  the  squares  of  two  lines 
is  equal  to  the  rectangle  of  the  sum  and 
difference  of  those  lines. 

Def.  76.  The  projection  of  a  point  on  a  straight 
line  is  the  foot  of  the  perpendicular  let  fall 
from  the  point  to  the  straight  line. 

Def.  77.  The  projection  of  a  finite  line  on  a 
straight  line  is  the  segment  cut  off  on  the 
latter  by  the  projection  of  the  extremities 
of  the  former. 

Prop.  93.  The  square  on  any  side  of  a  triangle  is 
equal  to  the  sum  of  the  squares  on  the  other 
two  sides  together  with  twice  the  rectangle 
formed  by  one  of  those  sides  with  the  pro- 
jection of  the  other  upon  it.  (By  the  use 
of  zero  and  negative  magnitudes.) 


38        SYLLABUS    OF   PLANE    GEOMETRY. 

Prop.  94.  The  sum  of  the  squares  on  any  two  sides 
of  a  triangle  is  equal  to  twice  the  square 
on  half  of  the  third  side,  increased  by  twice 
the  square  on  the  median  to  that  side. 

Prop.  95.  The  sum  of  the  squares  on  the  sides  of  a 
quadrilateral  is  equal  to  the  sum  of  the 
squares  on  the  diagonals  together  with  four 
times  the  square  of  the  lines  joining  the 
middle  points  of  the  diagonals. 


SECTION  3. 
The  Circle. 

Def.    78.        A  segment  of  a  circle  is  either  of  the  two 

figures  into  which  a  circle  is  divided  by  a 

chord. 
Def.    79.        The  segments  are  called  major  or  minor 

segments  according  as  their  arcs  are  major 

or  minor  arcs. 

Def.  80.  An  angle  is  said  to  be  inscribed  in  a 
circle  when  its  vertex  lies  on  the  circumfer- 
ence and  its  sides  form  chords  of  the  circle. 
An  inscribed  angle  is  said  to  stand  upon 
the  arc  between  its  arms.  An  inscribed 
angle  is  said  to  be  an  angle  in  that  segment 
whose  arc  is  the  conjugate  of  the  arc  upon 
which  it  stands. 


SYLLABUS    OF   PLANE    GEOMETRY.        39 

Prop.  96.  An  inscribed  angle  equals  half  the  angle 
at  the  center  standing  on  the  same  arc. 

Prop.  97.  Angles  in  the  same  segment  are  equal  to 
one  another. 

Prop.  98.  An  angle  in  a  segment  is  greater  than, 
equal  to,  or  less  than  a  right  angle,  accord- 
ing as  the  segment  is  less  than,  equal  to,  or 
greater  than  a  semicircle. 

Prop.  99.  The  opposite  angles  of  a  quadrilateral 
inscribed  in  a  circle  are  supplementary. 

Prop.  100.  Two  tangents,  and  only  two,  can  be  drawn 
to  a  circle  from  the  same  external  point. 

Prop.  101.  Tangents  to  a  circle  from  the  same  exter- 
nal point  are  equal. 

Prop.  102.  An  angle  formed  by  a  tangent  and  a 
chord  is  equal  to  half  the  angle  at  the 
center  standing  on  the  intercepted  arc. 

Prop.  103.  An  angle  formed  by  two  secants  inter- 
secting within  or  without  a  circle  is  equal 
to  half  the  sum  of  the  angles  at  the  center 
subtended  by  the  intercepted  arcs.  (By 
using  negative  quantities.) 

Prop.  104.  The  arcs  intercepted  on  a  circumference 
by  two  parallel  lines  are  equal. 

Prop.  105.  A  circle  can  be  circumscribed  about,  or 
inscribed  in,  a  regular  polygon. 

Prop.  106.  If  a  chord  of  a  circle  is  divided  into  two 
segments,  either  internally  or  externally, 
the  rectangle  contained  by  these  segments 


40        SYLLABUS    OF   PLANE    GEOMETRY. 

is  equal  to  the  difference  of  the  squares  on 
the  radius  and  on  the  line  joining  the  given 
point  with  the  center  of  the  circle. 

SECTION  4. 
Proportion. 

Theor.  O.  Kectangles  having  equal  altitudes  are  to 
one  another  as  their  bases. 

Cor.  Parallelograms  or  triangles  of  the  same 

altitude  are  to  one  another  as  their  bases. 

Prop.  107.  A  line  parallel  to  one  side  of  a  triangle 
divides  the  other  two  sides  proportionally. 
The  two  triangles  thus  formed  have  their 
corresponding  sides  proportional. 

Prop.  108.  A  line  can  be  divided,  internally  or  exter- 
nally, into  segments  having  a  given  ratio, 
except  that  if  it  is  divided  externally  the 
ratio  cannot  be  one  of  equality;  and  if  the 
line  is  given  in  both  length  and  sense,  the 
point  of  division  is  unique. 

Prop.  109.  The  line  which  divides  two  sides  of  a 
triangle  proportionally  is  parallel  to  the 
third  side. 

Theor.  P.  If  on  two  straight  lines,  cut  by  two  paral- 
lel straight  lines,  equimultiples  of  the  inter- 
cepts respectively  are  taken;  then  the  line 
joining  the  points  of  section  is  parallel  to 
either  of  the  two  parallels. 


SYLLABUS    OF   PLANE    GEOMETRY.        41 

Theor.  Q.  If  two  straight  lines  are  cut  by  three 
parallel  straight  lines,  the  intercepts  on  the 
one  are  proportional  to  the  corresponding 
intercepts  on  the  other. 

Oef.  81.  Two  figures  are  said  to  be  placed  in  per- 
spective when  the  straight  lines  joining 
corresponding  points  of  the  figures  pass 
through  a  common  point,  called  the  center 
of  perspective. 

Def.  82.  Two  figures  are  said  to  be  similar  when 
they  can  be  so  placed  that  the  center  of  per- 
spective divides  each  line  joining  corre- 
sponding points  into  segments  having  a  con- 
stant ratio. 

Def.  83.  The  center  of  perspective  is  then  called 
the  center  of  similitude  and  the  ratio  into 
which  it  divides  the  lines,  the  ratio  of 
similitude. 

Prop.  110.  Mutually  equiangular  triangles  are  simi- 
lar. 

Prop.  111.  If  two  triangles  have  an  angle  of  the  one 
equal  to  an  angle  of  the  other,  and  the  in- 
cluding sides  proportional,  they  are  similar. 

Prop.  112.  If  two  triangles  have  their  corresponding 
sides  proportional,  they  are  similar. 

Prop.  113.  Similar  triangles  have  their  correspond- 
ing angles  equal  and  their  corresponding 
sides  proportional. 

Prop.  114.       Similar  polygons  have  their  correspond- 


42        SYLLABUS    OF    PLANE    GEOMETRY. 

ing  angles  equal  and  their  corresponding 
sides  proportional. 

Prop.  115.  Similar  polygons  may  be  divided  into  the 
same  number  of  similar  triangles. 

Prop.  116.  Polygons  similar  to  the  same  polygon  are 
similar  to  each  other. 

Prop.  117.  If  in  a  right-angled  triangle,  a  perpendic- 
ular is  drawn  from  the  vertex  of  the  right 
angle  to  the  hypotemuse,  it  divides  the  tri- 
angle into  two  right  triangles  which  are 
similar  to  the  whole  triangle,  and  also  to 
each  other. 

Prop.  118.  Each  side  of  a  right-angled  triangle  is 
a  mean  proportional  between  the  hypote- 
muse  and  its  adjacent  segment. 

Prop.  119.  The  perpendicular  from  the  vertex  of  the 
right  angle  to  the  hypotemuse  of  a  right- 
angled  triangle  is  a  mean  proportional 
between  the  segments  of  the  hypotemuse. 

Prop.  120.  Similar  triangles  are  to  one  another  as 
the  squares  of  their  corresponding  sides. 

Prop.  121.  Similar  polygons  are  to  one  another  as 
the  squares  of  their  corresponding  sides. 

Prop.  122.  Any  polygon  described  on  the  hypote- 
muse of  a  right-angled  triangle  is  equal  to 
the  sum  of  the  two  similar  and  similarly 
described  polygons  on  the  sides. 


